DUDE! You're Double-Jointed!

There is no such thing as double jointed. It's just hyperflexibility and dexterity. You can doooooooo it. Here are some tips:
1. Stretch before and after you play.
2. Don't over-fret. Using too much pressure restricts your range of motion and also decreases dexterity. (Also decreases sustain and makes strings go sharp.) Practice playing single notes by using the least amount of fret hand pressure as possible while still allowing the string to sound properly. Lots of beginners correct for bad technique by clamping, as they (we - I did it too) improve, they forget to go back and review the fundamentals of fret hand pressure, pick hand pressure, etc. So make sure you're doing that.
3. See rule #1.
4. Practice moving chord shapes without playing (left hand only). What lots of players think of as poor mobility or poor flexibility is actually underdeveloped dexterity. Just practice fretting C/G/C/G/C/G over and over again without playing a song. Do that with all the important chord shapes you use on a daily basis.
5. Same thing goes for your fingers. Practice chord shapes using all fingers. Omit your index finger and try playing C, G, D with your middle, ring and pinky fingers. It sucks when you start but it gets easier.
6. As a last step, capo 2 or 3. Distance between frets gets shorter as you go up the neck. You might just need a little bit of help. Or play a gibson-scaled instrument.

-JL

Nothing wrong with blocking strings as long as you sound like you're trying to do it.